FCMLab: A Finite Cell Research Toolbox for MATLAB

N. Zandera,∗, T. Boga, M. Elhaddada, R. Espinozaa, H. Hua, A. Jolyb, C. Wua, P. Zerbea, A. Dusterc, S. Kollmannsbergera,

J. Parviziand, M. Ruesse, D. Schillingerf, E. Ranka

aChair for Computation in Engineering, Technische Universitat Munchen, Arcisstr. 21, 80333 Munchen, GermanybEcole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallee cedex 2, France

cNumerische Strukturanalyse mit Anwendungen in der Schiffstechnik, Technische Universitat Hamburg-Harburg,

Schwarzenbergstr. 95c, 21073 Hamburg, GermanydDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156 83111, Iran

eAerospace Structures and Computational Mechanics, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The NetherlandsfDepartment of Civil Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, 55455 MN, USA

Abstract

The recently introduced Finite Cell Method combines the fictitious domain idea with the benefits of high-order finite elements.

Although previous publications demonstrated the method’s excellent applicability in various contexts, the implementation of a

three-dimensional Finite Cell code is challenging. To lower the entry barrier, this work introduces the object-oriented MATLAB

toolbox FCMLab allowing for an easy start into this research field and for rapid prototyping of new algorithmic ideas. The paper

reviews the essentials of the methods applied and explains in detail the class structure of the framework. Furthermore, the usage of

the toolbox is discussed by means of different two- and three-dimensional examples demonstrating all important features of FCMLab

(http://fcmlab.cie.bgu.tum.de/).

Keywords: Finite Cell Method, fictitious domain methods, MATLAB, object-oriented finite elements, high-order finite elements,

p-FEM

1. Introduction

For decades, the Finite Element Method (FEM) has been

among the most prominent approaches to solve partial differ-

ential equations (PDEs) numerically. Many theoretical and al-

gorithmic enhancements of the method now allow to simulate

very complex scenarios in science and technology. What re-

mains unchanged, however, is the central idea to describe the

PDE’s solution and the domain geometry using the same mesh.

This dual-use of elements for the approximation of the geomet-

ric shape and solution characteristics implies e.g. that distorted

elements have to be avoided, as they would degrade the nu-

merical accuracy. For non-trivial geometries, this constraint

complicates the mesh generation processes considerably and

renders the method’s intrinsic need for a geometry-conforming

discretization as one often limiting factor in daily engineering

practice.

Alternative numerical approaches try to drop this limitation

by separating the discretization of the solution from that of the

geometry. Prominent examples are meshless and element-free

methods [1, 2]. Also the Generalized Finite Element, Immersed

Boundary, Fictitious Domain, Fat Boundary, Cartesian Grid-

FEM and certain variants of extended Finite Element Meth-

ods follow this approach [3–16]. For a comprehensive review

see [17–19]. A common idea of these methods is to approxi-

∗Corresponding author.

Email address: nils.zander@tum.de (N. Zander)

mate the PDE’s solution using a non geometry-conforming dis-

cretization (approximation mesh) and to capture the actual do-

main geometry on a separate geometry mesh. Many of these

approaches use a rather fine approximation mesh on which lin-

ear shape functions are spanned. This h-FEM-like concept

has been extended to higher orders by the Finite Cell Method

(FCM), combining a fictitious domain approach with the ben-

efits of high-order finite elements (p-FEM). The Finite Cell

Method was first introduced in [20, 21], where its potential was

demonstrated for linear-elastic examples in two and three di-

mensions. Various extensions of the FCM confirm its versatility

in the context of topology optimization [22], geometrically non-

linear continuum mechanics [23], adaptive mesh-refinement

[24–27], computational steering [28, 29], biomedical engi-

neering [30], numerical homogenization [31], elastoplasticity

[32], wave propagation in heterogeneous materials [33, 34],

local enrichment for material interfaces [35], convection dif-

fusion problems [36, 37], thin-walled structures [38], design-

through-analysis and iso-geometric-analysis [26, 27, 39, 40],

weak coupling of non-matching, multi-patch geometries [41],

and multi-physical applications [42]. The potential of high-

order-fictitious-domain methods is also demonstrated in the

context of Kantorovich methods [43, 44], immersed B-Spline

methods (I-Splines) [45, 6, 46], and eXtended Finite Element

Methods (XFEM) [13–16].

Yet, research in this field essentially demands for the avail-

ability of a high-order Finite Element code, which is not trivial

to implement. To lower this entry barrier, the goal of the present

Preprint submitted to Advances in Engineering Software May 23, 2014

work is to introduce FCMLab, an object-oriented MATLAB tool-

box tailored for quick prototyping of new algorithmic ideas.

Similar open-source software projects have been launched re-

cently in the communities of hp-FEM, XFEM, meshless meth-

ods, and isogeometric analysis (see e.g. [47–54]). The authors

hope that this work will be helpful for researches interested in

high-order fictitious domain methods. For this purpose, the sec-

ond section of this paper briefly explains the essential concepts

of high-order Finite Elements and of the Finite Cell Method.

The most important parts of the code design are outlined in the

third section. The fourth section explains how to use the tool-

box by different examples in two and three dimensions.

2. High-order Finite Elements and Finite Cells

As indicated in the introduction, this section outlines the

essential ideas of the Finite Cell Method. As a model prob-

lem, linear elasticity will be considered and some basics of the

p-version of the Finite Element Method, being a prerequisite

for the FCM will be sketched, briefly.

2.1. Model problem

Consider a two- or three dimensional domain Ω with bound-

ary ∂Ω divided into Neumann and Dirichlet parts ΓN and ΓD,

respectively, such that

ΓN ∩ ΓD = ∅ and ΓN ∪ ΓD = ∂Ω.

Assuming linear-elastic behaviour, the deformation u caused

by a body force b is described by the following set of partial

differential equations:

∇ · σ + b = 0 ∀ x ∈ Ω (1a)

σ = C : ε ∀ x ∈ Ω (1b)

ε =1

2

[

∇u + (∇u)⊤]

∀ x ∈ Ω (1c)

σ · n = t ∀ x ∈ ΓN (1d)

u = u ∀ x ∈ ΓD. (1e)

Here, σ and ε denote the stress and strain tensor, respectively,

and C is the linear-elastic constitutive tensor. t and u are the

traction and displacement vectors prescribed on the respective

boundaries. n is the outward-pointing normal vector on the

boundary.

Following the principal of virtual work described e.g. in [55–

58], the above equation is transferred into its corresponding

weak form:

Find u ∈ H1 (Ω) such that

a (u, v) = f (v) ∀ v ∈ H1 (Ω) (2a)

with a (u, v) =

∫

Ω

ε (v) : C : ε (u) dΩ (2b)

and f (v) =

∫

Ω

v · b dΩ +

∫

ΓN

v · t dΓ, (2c)

with H1 denoting the Sobolev space of order 1.

On the basis of the weak formulation, the Finite Element

Method is applied to approximate the analytical solution. To

this end, the infinite-dimensional Sobolev space is restricted to

an n-dimensional subspace Vh ⊂ H1 being spanned by the basis

B = N1,N2, . . . ,Nn. This allows to express the numerical ap-

proximation uh ∈ Vh as a linear combination of basis functions:

uh =

n∑

i=1

Niui, (3)

with ui denoting the degrees of freedom related to shape func-

tions Ni. Following a Bubnov-Galerkin approach, the space of

test functions v is reduced to the same space Vh [55]. Thus, the

same basis representation can be employed for the test functions

vh ∈ Vh. To determine the unknown degrees of freedom, the

above representation is inserted into the weak form (2) yielding

a system of linear equations

Ku = b, (4)

which has to be solved for the unknowns u. Typically, K and b

are denoted as stiffness matrix and external force vector.

2.2. p-version of the Finite Element Method

Nodal-Modes Edge-Modes

Edge-Modes Internal-Modes

N1Di

(r)

N1Dj

(s)

Figure 2: Tensor-product structure of legendre-based shape functions

The type of basis functions N has a major influence on the

characteristics of the numerical method. For the Finite Cell

Method, any high-order basis can be utilized. So far p-FEM

shape functions [57, 59], NURBS, being used in isogeometric

analysis [60], and Gauss-Lobatto-Legendre polynomials, used

in the Spectral Cell Method [33], have been applied success-

fully. Currently, FCMLab only provides basis functions from the

p-version.

2

-1 -0.5 0.5 1

-1

-0.5

0.5

1

(a) Linear modes

-1 -0.5 0.5 1

-1

-0.5

0.5

1

(b) Quadratic mode

-1 -0.5 0.5 1

-1

-0.5

0.5

1

(c) Cubic mode

Figure 1: 1D Legendre-based shape functions

-1

0

1

-1

0

1

0

1

2

(a) Nodal-mode

-1

0

1

-1

0

1

-1

-0.5

0

(b) Edge-mode

-1

0

1

-1

0

1

0

0.1

0.2

(c) Internal-mode

Figure 3: Two-dimensional mode types

Its one dimensional shape functions N1Di

(see Figure 1) are

defined as integrated Legendre polynomials [57]. They yield a

hierarchical basis, which is superior to the classical Lagrange

basis from a numerical point of view as the condition number

of the corresponding stiffness matrix remains constant for an in-

creasing polynomial degree (in case of Laplace problems). Al-

though this property does not carry over to higher dimensions,

a comparatively low condition number is maintained.

To obtain a two- or three-dimensional basis on a unit-square

or -cube, the one-dimensional shape functions are combined in

a tensor product

N2Di, j (r, s) = N1D

i (r) N1Dj (s)

and N3Di, j,k (r, s, t) = N2D

i, j (r, s) N1Dk (t) .

As depicted in Figure 2, the elements of the two-dimensional

tensor-product space can be sorted into the following three

groups:

Nodal-mode The combination of two one-dimensional nodal-

modes gives rise to the well known bi-linear modes (see

Figure 3a). Just as in 1D, these can be associated directly

to one node giving the group its name.

Edge-mode The combination of a one-dimensional internal-

mode and a one-dimensional nodal-mode results in a func-

tion that is non-zero at only one edge (see Figure 3b).

Therefore, the mode can be directly associated to that edge

and, thus, identified as a two-dimensional edge-mode.

Internal-mode The combination of two one-dimensional

internal-modes yields a function that is zero on all four

nodes and edges (see Figure 3c). The mode can, therefore,

be directly associated to the face and thus, identified as a

bubble- or internal-mode.

This mode concept naturally extends into three-dimensions,

where also internal- or volume-modes have to be considered.

Different alternatives to the tensor-product space, such as

a trunk-space or anisotropic polynomial-degree-templates, are

discussed in [61, 57]. They are currently not implemented in

FCMLab but might be added in the future.

2.3. The Finite Cell Method

As discussed in the introduction, mesh-generation is often

time consuming and sometimes even a limiting factor when us-

ing FEM for industrial applications. This problem becomes

even more severe for the p-version of the Finite Element

3

Ωphy

tt = 0 on ∂Ω∪

ΓN

ΓD

Ω f ic

Ω∪=Ωphy ∪ Ω f icα = 1.0

α = 0.0

Figure 4: The Finite Cell idea [23]

Method: although the solution can be represented by much less

and thus coarser elements compared to the h-version, they have

to assume possibly complex shapes of the domain’s surface, to

deliver accurate results [57]. The Finite Cell Method (FCM)

[20, 21], combining the benefits of higher-order Finite Elements

with the ideas of fictitious domain methods, overcomes these

mesh generation problems. The basics of the method are briefly

reviewed in the first part of this section, whereas the second

part addresses the imposition of boundary conditions for non

boundary-conforming discretizations.

2.3.1. Basic concept

The essential idea of the Finite Cell Method is to simplify the

mesh-generation process by separating the approximation of

the analytical solution from the geometry representation of the

physical domain. For this purpose, two independent discretiza-

tions are introduced: the solution and the integration mesh. The

first mesh is used to approximate the analytical solution and,

thus, spans the shape functions. However, it does not resolve

the geometry of the domain. This is done by the second mesh,

which is completely independent of the first one and does not

introduce any additional degrees of freedom.

Solution mesh. The first step to separate the two meshes is to

embed the actual physical domain Ωphy in a fictitious domain

Ω f ic such that their union Ω∪ yields a simple shape (see Fig-

ure 4). To recover the original boundary value problem on the

new domain Ω∪, an indicator function α is defined as

α(x) =

1 ∀ x ∈ Ωphy

0 ∀ x < Ωphy1.

(5)

1 Choosing α as zero outside of the physical domain negatively influences

the condition number of the stiffness matrix. To control the conditioning, the

value of α is typically set to a small value that allows to solve the system of

equations without numerical problems. Experience shows that the direct solvers

implemented in MATLAB can handle values of α ≈ 10−10 without major diffi-

culties.

The bi-linear form a(·, ·) can then be rewritten as

a (u, v) =

∫

Ωphy

ε (v) : C : ε (u) dΩ

=

∫

Ωphy

1 · ε (v) : C : ε (u) dΩ

+

∫

Ω f ic

0 · ε (v) : C : ε (u) dΩ

≈

∫

Ω∪

α · ε (v) : C : ε (u) dΩ.

The right hand side f (·) can be treated analogously. In this way,

the meshing of the complex physical domain can be simplified

drastically, as the simply-shaped domain Ω∪ can be discretized

instead using a Cartesian-shaped solution mesh, which renders

the mesh-generation process trivial. To avoid a confusion of

names, the resulting non boundary-conforming, high-order ele-

ments are denoted as Finite Cells giving the method its name.

Integration mesh. The second step is to recover the domain ge-

ometry using a separate integration mesh. In principle, this sec-

ond mesh can be generated following various strategies. One

possibility is to use conforming integration simplices by tri-

angulating the physical domain. XFEM- and Level Set-based

methods commonly apply this strategy [14, 15]. Typically, this

approach is used with linear simplices. However, the method

can also be extended to higher-orders as shown in [13, 62–

64, 16, 65, 66].

An alternative strategy, which is followed within this work,

is to use a space-tree as integration mesh (see e.g. [67]). Start-

ing from the non-conforming solution mesh, cut cells are recur-

sively refined towards the domain boundary yielding a quad- or

octree structure (see Figure 5). This approach has the advantage

that it requires only a simple inside-outside test. It can therefore

also be applied in conjunction with simple geometry modelling

techniques such as voxel-models, whereas the triangulation ap-

proach demands for a more advanced geometry representation.

2.3.2. Boundary conditions

A challenging task being a consequence of the non boundary-

conforming discretization is the imposition of boundary condi-

tions.

4

finite cell mesh

k=0 k=1 k=2

k=3 k=4 k=5

with geometric

boundary

Figure 5: Integration via space-trees [23]

Like in the classical FEM, homogeneous, ’natural’ Neu-

mann boundary conditions need no special treatment in the

FCM. Inhomogeneous Neumann boundary conditions demand

to explicitly include the boundary integral term of the linear

form (2c). As the boundary is not resolved by edges of the finite

cells, a classical integration over the element edges is not possi-

ble. Instead, an explicit surface discretization is introduced on

which the traction integral can be evaluated. Just as the previ-

ously described integration mesh, this surface integration mesh

is independent of the actual solution mesh and does not intro-

duce additional degrees of freedom. A convenient way to gener-

ate this mesh – supported by FCMLab– is to provide the surface

of interest as an STL2 file. The STL format is supported by

most CAD systems and represents surfaces as a collection of

linear triangles over which the integration can be performed.

In contrast to the Neumann case, the incorporation of Dirich-

let boundary conditions is more challenging. Classically, they

are imposed by explicitly choosing the shape functions from

kinematically admissible spaces, such that the prescribed val-

ues are met on the boundary by definition [57]. This imposes

the boundary conditions in the strong sense and can also be

followed in the case of non-boundary conforming discretiza-

tions by adapting the admissible function spaces for the partic-

ular geometry under consideration. Popular methods following

this idea are for example web- and i-spline based methods (see

e.g. [68, 45]). An alternative approach followed in this work

is to extend the weak formulation such that it directly incorpo-

rates the Dirichlet boundary conditions. Possible strategies for

this purpose are for example the Lagrange Multiplier or Penalty

Method [56, 69, 23, 70]. An alternative is to use an approach

originally introduced by Nitsche in [71] for the Laplace prob-

2Surface Tesselation Language

lem, which was later extended to other applications [72–75].

The idea of this method is to extend the original weak form (2)

by additional constraining expressions as follows:

a (u, v) = a (u, v) −

∫

ΓD

(σ (u) · n) · v dΓ

−

∫

ΓD

(σ (v) · n) · u dΓ + β

∫

ΓD

v · u dΓ

and f (v) = f (v) −

∫

ΓD

(σ (v) · n) · u dΓ + β

∫

ΓD

v · u dΓ.

The first additional integral of the bi-linear form follows natu-

rally from the divergence theorem when the test functions are

not restricted to be homogeneous on the boundary. It, therefore,

insures the method’s consistency. The second additional term

of a and the first additional term of f are introduced to yield a

symmetric stiffness matrix. The two remaining integrals are in-

troduced for numerical stabilization, since coercivity might be

lost when subtracting the consistency terms from the bilinear

form. With this extension, the discrete problem can be refor-

mulated incorporating the Dirichlet boundary conditions in the

weak sense as follows:

Find uh ∈V (Ω) such that

a (uh, vh) = f (vh) ∀ vh ∈ V (Ω) .

In analogy to Neumann boundary conditions, the constraining

expressions are evaluated using a separate surface integration

mesh.

The major advantage of Nitsche-like methods is their inher-

ent consistency, which ensures that the solution of the origi-

nal strong problem (1) also solves the modified weak problem

5

[73, 76, 74]. However, the major challenge of this approach is

to select the stability parameter β large enough to ensure the co-

ercivity of the modified weak form. As shown in [74, 72, 73],

the correct value of β is dependent on the mesh-size h, the poly-

nomial degree p, and the material parameters. [73–75] suggest

to solve an auxiliary eigenvalue-problem to estimate β. How-

ever, their investigations show that choosing a value higher than

the threshold has no significant effect on the numerical result.

Following the work of [23, 40, 42], who showed that also an

empirical choice of the stability parameter yields good results,

the eigenvalue-estimate is omitted in this work.

3. Code design

FCMLab is an object-oriented MATLAB toolbox, which is in-

tended to provide an easy entry point into the research field of

higher-order fictitious domain methods. This section presents a

compact outline of the code design. To this end, the two most

frequently applied design patterns are introduced in the first part

of the section. In the second part, the most important classes of

the toolkit and their mutual dependencies are explained. The

description of the code expects the reader to be familiar with

general MATLAB programming and knowledge of object ori-

ented concepts in MATLAB, as described e.g. in [78, 79].

3.1. Applied design patterns

In the context of object-oriented software engineering, cer-

tain types of design problems keep on re-appearing, indepen-

dent of the actual application under consideration. For this rea-

son, design patterns have been developed as reusable template

solutions for the most prominent design problems. A classi-

cal reference in this context is the textbook by Gamma, Helm,

Johnson, and Vlissides [77].

3.1.1. Strategy pattern

The strategy pattern is a behavioral pattern aiming to decou-

ple a family of algorithms from the actually calling code. For

this purpose, different algorithms are implemented as separate

classes. They all inherit from the same base class which defines

the common interface. The client code then implements against

this abstract interface and not against the actual implementa-

tions. Thus, the calling codes and the algorithms are decoupled,

allowing to refactor and maintain the implementation of differ-

ent algorithms without affecting the client code. Furthermore,

the family of algorithms can be extended by simply adding new

sub-classes. A numeric-oriented example is depicted in Fig-

ure 6a.

3.1.2. Factory method pattern

The factory method pattern is a creational pattern aiming to

simplify the creation of complex products. For this purpose,

the creational process is decoupled from the client code by

introducing an abstract creator class. This specifies the inter-

face functions that create the respective object. These factory-

methods are then implemented in different sub-classes deciding

in which way the final product is constructed. This approach

allows the client code to implement against the abstract inter-

face without knowing about the details of the different creator

classes. In this way, the implementation of the creational pro-

cess can be exchanged easily as the calling code remains unaf-

fected. A numeric-oriented example is depicted in Figure 6b.

The factory idea can be extended to the abstract factory pat-

tern, which allows to create complex product families easily.

Within FCMLab, both versions of the pattern are applied and

combined.

3.2. Code Verification

Besides the aspects of extensibility and maintainability, a

third important request when developing a software framework

is to ensure and maintain code correctness and consistency. For

this purpose, FCMLab is developed in a test-driven manner (see

e.g. [80, 81]). The framework is, thus, equipped with a test

suite, which verifies the correctness of the individual modules

following the idea of unit tests. Furthermore, the full simula-

tion pipeline is tested using examples with analytical solutions

which allow to verify the numerical results. To ensure code

correctness during the on-going development process, a contin-

uous integration system is employed, which automatically exe-

cutes the entire test suite as soon as changes are committed to

the version control system. To this end, we utilize the MAT-

LAB xUnit Test Framework [82], the continuous integration

server Jenkins (formally known as Hudson) [83], and Apache

Subversion (SVN) [84].

3.3. Program structure

The aim of FCMLab is to provide a prototyping framework in

which new algorithmic ideas can be tested easily. Therefore,

the framework has to be structured such that different com-

ponents can be exchanged easily without affecting other parts

of the code. The design must thus aim for high cohesion of

the individual modules – in the sense that all libraries, classes,

and functions have one clearly defined responsibility–and low

coupling between the modules. To address these issues, the

program is decomposed hierarchically into the sub-systems de-

picted in Figure 7. These are then arranged as layers such that

each module only depends on lower layers. This architecture

Geometry

Embedded Domain

Integration

Topology

Material

Element

Mesh

Boundary Conditions

Analysis Post Processing

Driver Files

Figure 7: Layered decomposition of FCMLab

6

MultiGrid GaussSeidelCG

Analysis

+run(...)

AbsSolver

+solve(...)

(a) Strategy Pattern

1DMeshFactory 2DMeshFactory

AbsMeshFactory

+createElements(...)

+createNodes(...)

ElasticQuad

AbsElement

Mesh

ElasticBar

Topology

(b) Factory Method Pattern

Figure 6: Commonly used design patterns [77]

allows to regard each layer as a virtual machine with a clearly

defined task, whose implementation can be exchanged without

affecting up-stream elements. The details of the different pack-

ages are explained in the following sections.

3.3.1. Geometry

The geometry packages provide essential geometric objects

such as vertices, lines, quadrilaterals, and hexahedra (see Fig-

ure 8). These are realized as separate classes and grouped ac-

cording to their dimensionality as abstract curves, areas, and

volumes. These abstractions all implement the class AbsGeom-

etry that specifies the interface functions listed in Table 1.

Function Name Parameter

calcJacobian(...) local coordinates

mapLocalToGlobal(...) local coordinates

mapGlobalToLocal(...) global coordinates

Table 1: Interface-functions of geometry package

TriQuad

AbsCurve

Hexa

Vertex AbsVolumeAbsArea

Line

AbsGeometry

+calcJacobian(...)+mapLocalToGlobal(...)

+mapGlobalToLocal(...)

Figure 8: Class structure of geometry package

3.3.2. Topology

As discussed in Section 2.2, the Legendre-based shape func-

tions can be associated to the different topological components.

Therefore, the topology package, depicted in Figure 9, is re-

sponsible for handling the degrees of freedom. To this end,

a Dof class is defined, which is then aggregated by the Abs-

Topology class. This interface is implemented by the classes

Node, Edge, Face, and Solid that hold objects of their respec-

tive geometric counterparts. Furthermore, the classes store the

associated polynomial degree of the shape functions.

3.3.3. Embedded domain

As discussed in Section 2.3, the essential idea of the Finite

Cell Method is to decouple the solution mesh from the geome-

try and to recover the original domain on the integration level.

This demands for a separate geometry representation, which is

provided by the embedded domain package (see Figure 10). It

contains the abstract AbsEmbeddedDomain class, which spec-

ifies the interface function getDomainIndex(...) determin-

ing the domain in which a point under consideration lies. This

interface is implemented by different concrete domains, such as

simple geometric objects that can be specified explicitly (e.g.

sphere, rectangle, ellipse). Furthermore, FCMLab provides an

implementation that allows to create embedded domains on the

basis of CT-scan-data. This VoxelDomain class reads in the

CT-data from a single ASCII file format, which linearizes the

three-dimensional matrix. The origin and the bounding box are

extracted automatically. Based on this information, the Finite

Cell mesh can be created. The list of domain types can be ex-

tended easily to suit new needs.

In addition to the domain description, the FCM demands for

an explicit representation of the boundary for application of

boundary conditions. For this purpose, the package provides a

AbsBoundaryFactory class, which specifies the getBound-

ary(...) function returning a surface description as a list of

geometric simplices. Presently, several explicit geometries such

as circles and rectangles are provided as well as an implemen-

tation to read STL files3, which can be exported from a CAD

model. Again, this list can be extended easily for new applica-

tions.

3For this purpose, the stlread function provided by Doron Harlev is used,

which is available on the MATLAB Central File Exchange platform: http://

www.mathworks.com/matlabcentral/fileexchange/6678-stlread

7

3.3.4. Integration

As discussed in Section 2.3, the domain integration for the

FCM is carried out over a separate integration mesh. For this

purpose, the integration package defines the non-abstract class

Integrator. This class provides the service function in-

tegrate(...), taking the integrand and the integration do-

main as arguments. To generate the integration mesh, the in-

tegrator aggregates the abstract classes AbsPartitioner and

AbsIntegrationScheme. The AbsPartitioner serves as an

interface for different partitioning strategies (see Figure 11).

Currently, FCMLab provides one-, two-, and three-dimensional

space trees as concrete partitioning strategies. These could be

extended by different schemes, such as triangulations. The

class AbsIntegrationScheme provides interfaces to get one-

dimensional integration coordinates and weights. At present,

the quadrature rule for Gauss-Legendre integration is available,

which is used mostly in Finite Element Analysis [55].

3.3.5. Material

The material package contains different constitutive rela-

tions. To encapsulate these from the other parts of the code,

the AbsMaterial class is defined, which specifies the interface

functionality listed in Table 2.

In addition to providing the constitutive properties, the mate-

rial package is also responsible for defining the Finite Cell indi-

cator function α introduced in Section 2.3. For this purpose, the

third method in the table is specified, which returns the α-value

of the respective material. The idea is to associate one material

for each domain: on the physical domains, the “real” materials

are defined with α = 1.0; on the fictitious domains, auxiliary

“void” materials are defined with α→ 0 4. This concept allows

to handle all domains and materials in the same way during the

integration process.

As depicted in Figure 12, FCMLab provides material laws for

linear-elastic simulations. Again, this list can be extended by

new materials easily as the other parts of the code implement

against the interface defined by AbsMaterial.

4To avoid numerical difficulties due to badly conditioned matrices, an α-

value of 10−10 is chosen for the numerical examples of Section 4

Node Solid

AbsTopology

+getDofs(...)

Dof

+getId(...)

+getValue(...)

+setValue(...)

Vertex

Face

AbsCurve

Edge

AbsArea AbsVolume

Figure 9: Class structure of topology package

VoxelDomainAnnularPlate IBeam

AbsEmbeddedDomain

+getDomainIndex(...)

(a) Selection of provided domains

CircularBoundaryFactory

AbsBoundaryFactory

+getBoundary(...)

STLBoundaryFactory

(b) Selection of provided boundaries

Figure 10: Class structure of embedded domain package

AbsEmbeddedDomain

AbsPartitioner

+subdivide(...)

OctTree GaussLegendre

AbsIntegrationScheme

+getCoordinates(...)

+getWeights(...)

QuadTree

AbsGeometry

AbsIntegrator

+integrate(...)

Integrand

Figure 11: Class structure of integration package

Function Name Parameter

getMaterialMatrix(...) none

getDensity(...) none

getScalingFactor(...) none

Table 2: Interface-functions of material package

3.3.6. Element

The element package is the kernel of the toolbox. It is re-

sponsible for computing the stiffness matrix and the right-hand-

side vector of an element. For this purpose, the abstract Abs-

Element class is defined aggregating its topological compo-

nents, a material, and a domain and offers the interface func-

tions listed in Table 3. To implement these services, the shape

functions and their derivatives need to be evaluated and ar-

ranged in matrices. As this process depends on the dimension-

ality of the problem, the strategy pattern is applied by encapsu-

lating the shape-function evaluation into respective sub-classes

(see Figure 13). In this way, the higher-level parts of the code

8

HookePlaneStress HookePlaneStrain

AbsMaterial

+getMaterialMatrix(...)

+getDensity(...)

+getScalingFactor(...)

Hooke3DHooke1D

Figure 12: Class structure of material package

can use the abstract element without having to know about the

details of the setup and the dimensionality. As the creation of

an element is rather complex, different factories are provided to

generate the most commonly used types.

Function Name Parameter

calcStiffnessMatrix(...) none

calcMassMatrix(...) none

calcBodyLoadVector(...) load function

getLocationMatrix(...) none

getDofVector(...) load case id

Table 3: Interface-functions of element package

ElasticHexaElasticQuad

AbsElement

+calcStiffnessMatrix(...)

+calcMassMatrix(...)+calcBodyLoadVector(...)

+getLocationMatrix(...)

+getDofVector(...)

AbsMaterial

ElasticBar

Figure 13: Class structure of element package

3.3.7. Mesh

The mesh package provides components to create and oper-

ate on one-, two-, and three-dimensional Cartesian meshes. For

this purpose, a non-abstract Mesh class is defined aggregating

the respective topological components as well as all elements

(see Figure 14). As services, it offers the functions listed in Ta-

ble 4. Furthermore, the mesh package offers different schemes

to number the degrees of freedom. To this end, the AbsNumber-

ingScheme is defined, which assigns the degrees of freedoms

to the topological components. The interface is implemented

by concrete strategies sorting the unknowns either according to

their associated polynomial degree or to their topological com-

ponent.

Following the abstract factory pattern, the mesh creation pro-

cess is encapsulated into the abstract AbsMeshFactory class

that handles the setup of the topology, the elements, and the

numbering scheme.

AbsNumberingScheme

+assignDofs(...)

TopologicalSorting

AbsElement

Mesh

+assembleStiffnessMatrix(...)

+assembleMassMatrix(...)+calcBodyLoadVector(...)

+assembleLoadVector(...)

+scatterSolution(...)

PolynomialDegreeSorting

Figure 14: Class structure of mesh package

Function Name Parameter

assembleStiffnessMatrix(...) none

assembleMassMatrix(...) none

assembleLoadVector(...) load case id

scatterSolution(...) solution

Table 4: Interface-functions of mesh package

3.3.8. Boundary condition

The boundary condition package offers the possibility to

constrain the numerical system using Neumann and Dirichlet

boundary conditions. As described in Section 2.3, the latter can

be applied in the strong and in the weak sense. To break the

complexity of the package, it is split into sub-systems, which

are explained in the following paragraphs5.

Dirichlet boundary conditions. To apply Dirichlet boundary

conditions, the abstract AbsDirichletBoundaryCondition

class is introduced. It provides the interface function modi-

fyLinearSystem(...), which takes the mesh and the sys-

tem of linear equations as arguments and applies the bound-

ary conditions. From this super class, the StrongDirichlet-

BoundaryCondition and WeakDirichletBoundaryCondi-

tion are derived.

Strong Dirichlet boundary conditions directly operate on

degrees of freedom. Since the latter are associated to

5Within the figures of the different packages, the terms DBC and NBC are

used as abbreviations for Dirichlet and Neumann boundary conditions. Within

FCMLab, the full class names are spelled out.

9

StrongPenaltyDirichletAlgorithm

AbsStrongConstrainingAlgorithm

+modifyLinearSystem(...)

AbsStrongDBC

+modifyLinearSystem(...)

StrongDirectConstrainingAlgorithm

StrongFaceDBCStrongEdgeDBCStrongNodeDBC

Figure 15: Class structure of strong boundary condition Package5

WeakPenaltyAlgorithm WeakNitscheAlgorithm

WeakDBC

+modifyLinearSystem(...)

AbsWeakConstrainingAlgorithm

+modifyLinearSystem(...)

Integrator

AbsTractionOperator

+getTractionVector(...)

TractionElastic3DTractionElastic2DAbsGeometry

Figure 16: Class structure of weak boundary condition Package5

LoadFunction

LoadCase

+addBodyLoad(...)

+addNeumannBoundaryCondition(...)

WeakNeumannBoundaryCondition

AbsIntegrationScheme

NodalNeumannBoundaryCondition

AbsNeumannBoundaryCondition

+augmentLoadVector(...)

AbsGeometry

Figure 17: Class structure of neumann boundary condition Package5

10

the topological components (see Section 3.3.2), three differ-

ent classes are derived from AbsStrongDirichletBound-

aryCondition. They handle the corresponding tasks for

nodes, edges, and faces, respectively (see Figure 15). In prin-

ciple, strong Dirichlet boundary conditions can be applied in

various ways. To allow for this flexibility, the strategy pat-

tern is applied by introducing the interface class AbsStrong-

ConstrainingAlgorithm. Presently, two implementations

of this interface are provided: The first is the StrongPenal-

tyDirichletAlgorithm, which constrains the respective de-

grees of freedom by adding a specified penalty value on the

main diagonal of the system matrix. The second strategy con-

denses the influence of the constrained degrees of freedom to

the right hand side and thus results in a better conditioning of

the stiffness matrix (see e.g. [85]).

As described in Section 2.3, weak Dirichlet boundary con-

ditions can be e.g. applied following the penalty or a Nitsche-

like method. Both approaches introduce additional boundary

integrals constraining the solution appropriately. To evalu-

ate these integrals, the WeakDirichletBoundaryCondition

class aggregates a list of geometric objects as boundary de-

scription and an integrator object (see Figure 16). The ac-

tual constraining algorithm is encapsulated by introducing

the AbsWeakConstrainingAlgorithm interface class, which

is then implemented by the WeakNitscheAlgorithm and

WeakPenaltyAlgorithm.

Neumann boundary conditions. Also in the case of Neumann

boundary conditions, two different cases have to be considered.

In the general case, the traction integral in (2c) has to be eval-

uated over the Neumann boundary. To this end, the WeakNeu-

mannBoundaryCondition class is introduced, which, in anal-

ogy to the weak Dirichlet case, aggregates a boundary descrip-

tion and an integrator object. The actual integrand is specified

as a handle to a load function (see Figure 17).

The second case is the application of a load vector, which

lumps the applied tractions in the nodal degrees of freedom

[58]. For this purpose, the NodalNeumannBoundaryCondi-

tion is defined.

To compute the mechanical reactions of the system under

consideration for different load settings, a LoadCase class is

introduced to which different body and surface loads can be as-

sociated.

3.3.9. Analysis

The analysis package is at the top level of the FCMLab tool-

box. Here the numerical problem is set up and run. For this

purpose, the abstract class AbsAnalysis is introduced, which

aggregates the mesh and boundary conditions (see Figure 18)

and specifies the interface functions listed in Table 5. These

are then implemented by concrete analyses. In its current state

FCMLab provides a quasi-static and an eigenmode analysis (see

Figure 18).

3.3.10. Post-processing

The final module to be discussed here is the post-processing

package. FCMLab offers two different ways to post-process nu-

EigenmodeAnalysis

LoadCaseMesh

QuasiStaticAnalysis

AbsAnalysis

+addLoadCases(...)+addDirichletBoundaryCondition(...)

+solve(...)

AbsDrichletBC

Figure 18: Class structure of analysis package5

Function Name Parameter

addLoadCases(...) list of load cases

addDBC(...) abstract Dirichlet BC

solve(...) none

Table 5: Interface-functions of analysis package

merical results: visual post-processing depicting the results as

line or surface plots and integrational post-processing allow-

ing the calculation of integral values, such as strain energy and

mass. Again following the strategy pattern, both post-processor

types operate on point-processors via an interface class Abs-

PointProcessor. These convert the numerical solution into

the physical quantities of interest, such as strains, stresses, and

energies (see Figure 19).

In the context of the Finite Cell Method, both post-processing

types have to be enhanced to operate on the non boundary-

conforming discretization. In case of integrational post-

processing, the quadrature package outlined in Section 3.3.4

is used for this purpose. For visual post-processing, a sepa-

rate post-processing mesh is introduced to visualize the results

in two or three dimensions. Different factories provided by

FCMLab allow for an easy creation of these meshes. The user

can also choose between different types of visualization, such

as line- or path- and surface-plots. For the latter plot type, also

the finite cells, the integration cells, and the boundary condition

mesh can be visualized.

3.4. Extensions to alternative differential operators and non-

linear problems

Currently the toolbox only supports simulations of linear

elastic problems. As shown by e.g. Schillinger et al., however,

the Finite Cell Method also yields excellent results in the con-

text of geometrically non-linear elasticity (see e.g [27]). The

modular setup of FCMLab, described in the previous sections,

allows to easily extend the toolbox for applications of this cat-

egory. As a guideline for users of the toolbox, the necessary

steps shall be briefly outlined in the following.

First, new elements have to be defined, which allow for large

deformations. To this end, new element classes have to be cre-

ated in the element package, which implement the AbsElement

11

VonMisesStress

SurfacePlot IntegrationMeshVisualizer

IntegrationPostProcessor

+integrate(...)

StrainEngery

Triangulation3D

AbsPostProcessor

+registerPointProcessor(...)

Triangulation2DStraightLine

VisualPostProcessor

+visualizeResults(...)

AbsPostProcessingMeshFactory

+createPostProcessingMesh(...)

AbsVisualizer

+visualizeResults(...)

L2ErrorDisplacement

LinePlot

PostProcessingMesh

AbsPointProcessor

+evaluate(...)

Figure 19: Class Structure of Post-Processing Package. Note that only a selection of the available point-processors, mesh factories, and visualizers is depicted.

interface.

In a second step, the respective material models (e.g. Moony-

Rivlin or Neo-Hooke [86]) have to be implemented in the

framework. This can be done easily by creating new sub-classes

in the material package that implement the AbsMaterial inter-

face.

The final step is the implementation of a non-linear solver,

which solves the non-linear system of equations iteratively e.g.

using the Newton’s method or a fix-point approach (see e.g.

[86]). This has to be carried out in the analysis package by

creating new classes, which implement the AbsAnalysis in-

terface.

Similarly, the framework could be extended into the domain

of fluid dynamics, where void and structure essentially inter-

change their role, see, e.g. [87]. For a convection diffusion

problem the penalization factor α is applied to the diffusion co-

efficient in the ’obstacle’ regime of the flow.

4. Examples

Having outlined the theoretical background and the basic

program structure in the previous sections, this part of the work

aims at demonstrating the usage of FCMLab. For this purpose,

examples from previous FCM-related publications are reused

for highlighting the different features of the toolbox. The sec-

tion starts with a two-dimensional benchmark example with an

explicit geometry description. In the second example, the me-

chanical behaviour of a three-dimensional femur (thigh bone)

is investigated using a voxel model. The section concludes

with a third example demonstrating the possibility of eigen-

mode analyses using the Finite Cell Method.

4.1. Analytical benchmark

This first example outlines the structure of a driver file to

setup an analysis within FCMLab. For this purpose, the dis-

placement of the annular plate depicted in Figure 20a shall be

computed, following [23, 42, 40]. Assuming a plane stress con-

figuration and the denoted material properties, the displacement

in polar coordinates caused by the stated radial loading can be

given analytically as:

ur = −r

2

ln r

ln 2

uθ = 0.

The corresponding polar stresses read

σr = εr =∂ur

∂r= −

1

2

ln(r) + 1

ln(2)

σθ = εθ =1

r

∂uθ

∂θ+

ur

r= −

1

2

ln(r)

ln(2)

σrθ = εrθ =1

2

(

1

r

∂ur

∂θ+∂uθ

∂r−

uθ

r

)

= 0.

The analysis can be separated into the parameter configura-

tion and the analysis setup. These two sections are explained

individually in the following paragraphs.

Parameter configuration. The driver file starts with the de-

scription of the discretization and material parameters:

1 MeshOrigin = [-1.1 -1.1 0.0];

2 MeshLengthX = 2.2;3 MeshLengthY = 2.2;4 NumberOfXDivisions = 4;

5 NumberOfYDivisions = 4;6 PolynomialDegree = 4;

7 NumberOfGaussPoints = PolynomialDegree+1;

8 YoungsModulus = 1.0;9 PoissonsRatio = 0.0;

10 Density = 1.0;

In a second step, the numerical parameters of the Finite Cell

Method are specified as follows:

11 SpaceTreeDepth = 3;

12 Alpha = 1e-10;13 Beta = 1e3;

Here the value of the indicator function α outside of the physi-

cal domain is chosen to be a small positive number in order to

avoid ill-conditioning of the system matrix.

12

The third step is to define the inner and outer radius of the

annular plate and the discretization of the boundary mesh:

14 Center = [ 0.0 0.0 0.0 ];15 OuterRadius = 1.0;

16 InnerRadius = 0.25;17 NumberOfSegments = 500;

Finally, the body forces and the boundary values need to be

specified as function handles:

18 bPolar = @(r,theta) [ 1/(r*log(2)) ; 0];

19 uxPolar = @(r,theta) 0.25* cos(theta);

20 uyPolar = @(r,theta) 0.25* sin(theta);

Analysis setup. After having specified the different numerical

parameters, the analysis can be setup. For this purpose, two

different materials are defined for the physical and the void do-

main:

21 Materials(1) = HookePlaneStress( YoungsModulus , ...22 PoissonsRatio , Density , Alpha );

23 Materials(2) = HookePlaneStress( YoungsModulus , ...24 PoissonsRatio , Density , 1 );

The next step is to define the physical domain and its bound-

ary. As the geometry can be described explicitly, the Annu-

larPlate implementation of the AbsDomain interface intro-

duced in Section 3.3.3 is used:

25 AnnularPlate = AnnularPlate( ...26 Center , OuterRadius , InnerRadius );

27 CounterClockWise = true;28 OuterBoundaryFactory = ...

29 CircularBoundaryFactory( Center, ...30 OuterRadius , NumberOfSegments , ...

31 CounterClockWise );

32 CounterClockWise = false;

33 InnerBoundaryFactory = ...34 CircularBoundaryFactory( Center, ...

35 InnerRadius , NumberOfSegments , ...36 CounterClockWise );

In a third step, the discretization is created using the element

and mesh factories described in Section 3.3.6 and 3.3.7:

37 DofDimension = 2;

38 ElementFactory = ...

39 ElementFactoryElasticQuadFCM( ...40 Materials , AnnularPlate , ...41 NumberOfGaussPoints, SpaceTreeDepth );

42 MeshFactory = MeshFactory2DUniform( ...

43 NumberOfXDivisions, NumberOfYDivisions ,...44 PolynomialDegree , PolynomialDegreeSorting, ...

45 DofDimension , MeshOrigin , MeshLengthX , ...46 MeshLengthY , ElementFactory );

With these entities, the quasi-static analysis can now be cre-

ated:

48 Analysis = QuasiStaticAnalysis( MeshFactory );

To apply the body forces, a new load case is set up and reg-

istered using the function handles defined above:

49 BodyForce = @(x,y,z) ...

50 polarToCartesian2DVector( x, y, bPolar );

51 LoadCase = LoadCase ();52 LoadCase .addBodyLoad( BodyForce );

53 Analysis .addLoadCases( LoadCase );

To specify the weak Dirichlet boundary conditions, a Nitsche

constraining algorithm and an integration scheme are specified.

These are then used to create three boundary conditions for the

inner and outer boundary in x- and y-direction, which are then

registered to the analysis:

54 % Select constraining strategy

55 ConstrainingAlgorithm = ...56 WeakNitscheDirichlet2DAlgorithm( Beta );

57 IntegrationScheme = ...58 GaussLegendre( NumberOfGaussPoints );

59 % Create outer boundary condition60 Boundary = ...

61 OuterBoundaryFactory.getBoundary();62 zeroBoundary = @(x,y,z) 0;63 ConstrainedDirections = [1 1];

64 OuterCondition = WeakDirichletBoundaryCondition( ...65 zeroBoundary , ConstrainedDirections, ...

66 IntegrationScheme, Boundary , ...67 ConstrainingAlgorithm );

68 % Create inner boundary condition in X69 Boundary = ...

70 InnerBoundaryFactory.getBoundary();71 Ux = @(x,y,z) polarToCartesian(x,y,uxPolar );

72 ConstrainedDirections = [1 0];73 InnerConditionX = WeakDirichletBoundaryCondition( ...74 Ux, ConstrainedDirections, IntegrationScheme ,...

75 Boundary , ConstrainingAlgorithm );

76 % Create inner boundary condition in Y77 Uy = @(x,y,z) polarToCartesian(x,y,uyPolar );

78 ConstrainedDirections = [0 1];79 InnerConditionY = WeakDirichletBoundaryCondition( ...80 Uy, ConstrainedDirections, IntegrationScheme ,...

81 Boundary , ConstrainingAlgorithm );

82 % Register boundary conditions at analysis83 Analysis .addDirichletBoundaryCondition( ...84 OuterCondition );

85 Analysis .addDirichletBoundaryCondition( ...86 InnerConditionX );

87 Analysis .addDirichletBoundaryCondition( ...88 InnerConditionY );

With this setup, the analysis can be executed as follows:

89 Analysis .solve();

As discussed in Section 3.3.10, FCMLab offers the possibli-

ties of visual and integral post-processing, which both work on

point-processors. These can, for example, be defined as fol-

lows:

90 indexOfPhysicalDomain = 2;91 loadCaseToVisualize = 1;

92 displacementNorm = DisplacementNorm( ...93 loadCaseToVisualize );

94 vonMises = VonMisesStress( loadCaseToVisualize, ...95 indexOfPhysicalDomain );96 strainEnergy = StrainEnergy( loadCaseToVisualize, ...

97 indexOfPhysicalDomain );

To visualize these results, a visual post-processor is created

to which different point-processors are registered:

98 FeMesh = Analysis .getMesh ();99 gridSize = 0.1;

100 postProcessingFactory = ...101 VisualPostProcessingFactory2DFCM( ...102 FeMesh, AnnularPlate , ...

103 indexOfPhysicalDomain, gridSize , ...104 OuterBoundaryFactory, ...

105 InnerBoundaryFactory );106 postProcessor = ...

107 postProcessingFactory.creatVisualPostProcessor();

13

x

y

r1 = 0.25

r2 = 1

L = 2.2

ur |r2= 0

ur |r1= 0.25

E = 1ν = 0

br =1

r ln 2

Ωphys

Ω f ict

(a) Benchmark setup

0.05 0.1 0.15 0.2 0.25

(b) Displacement norm

0.5 0.6 0.7 0.8 0.9

(c) von Mises stress

Figure 20: Visualization of annular plate results

108 postProcessor.registerPointProcessor( ...109 displacementNorm );

110 postProcessor.registerPointProcessor( vonMises );

111 postProcessor.visualizeResults( FeMesh );

The visualization of the displacement and stress obtained from

the registered post-processors is depicted in Figure 20.

In analogy, the results can also be integrated over the domain:

112 postProcessor = IntegrationPostProcessos( );

113 postProcessor.registerPointProcessor( strainEnergy );

114 postProcessor.registerPointProcessor( l2Error );

115 integrals = postProcessor.integrate( FeMesh );

4.2. Femur

The following example aims at demonstrating the ability of

FCMLab to work on geometries defined by voxel data. To this

end, the mechanical reaction of a human femur (thigh bone)

subjected to a hip-contact force is computed. Typically, inho-

mogeneous material models are applied in this context to cap-

ture the physical behaviour correctly (see e.g. [30, 88, 89]).

However, as the focus of this paper lies on demonstrating the

usage of different geometry models, a homogeneous, linear-

elastic material description is used.

As the analysis is again quasi-static, the general setup of the

driver file remains unchanged. However, an important differ-

ence is that the bone’s geometry is represented by a voxel-

model obtained from a CT6-scan of the patient. The data file

is loaded using the VoxelDomain implementation of the Abs-

Domain interface described in Section 3.3.3:

1 VoxelDataFile = ’BoneFF1_CTdata_cut.txt’;2 Bone = VoxelDomain( VoxelDataFile );

The second important difference is that the hip-contact force has

to be applied on the non boundary-conforming discretization.

To this end, a separate surface discretization of the hip-cap is

loaded from an STL-file and then used to create a Neumann

boundary condition:

6Computed Tomography

−30−20−10

010

0

5

10

15

20

25

30

−60

−40

−20

0

200

50

100

150

Figure 21: von Mises stress distribution in MPa

14

1 BinarySTLFile = ’HipCap.stl’;

2 LoadSurfaceFactory = ...3 STLBoundaryFactory( BinarySTLFile );

4 LoadBoundary = ...5 LoadSurfaceFactory.getBoundary();

6 BoundaryIntegrator = ...7 GaussLegendre( NumberOfGaussPoints );

8 pressure = -1;

9 Traction = @(x,y,z) [0;0; pressure ];10 Load = WeakNeumannBoundaryCondition( ...11 Traction , BoundaryIntegrator, ...

12 LoadBoundary );

13 LoadCase = LoadCase ();14 LoadCase .addNeumannBoundaryCondition( Load );15 Analysis .addLoadCases( LoadCase );

The remaining parts of the driver file follow in analogy to the

first example and are, thus, not recapitulated here.

Of course, also three-dimensional results can be post-

processed using FCMLab. Figure 21 depicts exemplarily the

von Mises’ stress distribution in the bone caused by the load-

ing. The figure also shows the traction surface used to apply the

hip-contact forces on the bones’ head.

4.3. Eigenmode analysis

This final example aims at demonstrating the use of FCMLab

in the context of an eigenmode analysis. For this purpose,

eigenmodes of the I-section beam depicted in Figure 22 are

computed.

In line with the annular plate example, the physical domain

is represented by an explicit geometry description. As shown in

Figure 22a, the beam is embedded in a slightly larger fictitious

domain being discretized by 5× 5× 5 finite cells of polynomial

order 3. The driver file for this setup follows in analogy to the

previous two examples. The important difference is that in this

application no quasi-static analysis is performed. Instead, the

EigenmodeAnalysis implementation of the AbsAnalysis in-

terface is applied in the following way:

1 NumberOfModesToCompute = 10;2 Analysis = EigenmodeAnalysis( ...

3 MeshFactory , NumberOfModesToCompute );

Results are depicted in Figure 23, and the corresponding fre-

quencies are listed in Table 6. The comparison to references

obtained from an ANSYS computation using 3200 quadratic

elements of type Solid186 (see [90]) demonstrates that the Fi-

nite Cell Method does not only allow to significantly simplify

the mesh generation process but also yields highly accurate re-

sults.

5. Conclusion

This work introduced FCMLab as an object-oriented MAT-

LAB toolbox for the Finite Cell Method allowing rapid-

prototyping of new algorithmic methods in the context of high-

order fictitious domain methods.

7Degrees of Freedom

FCMLab ANSYS Deviation

1st Frequency 7.308 7.307 1.37e-02 %

2nd Frequency 9.637 9.635 2.08e-02 %

3rd Frequency 33.452 33.325 3.81e-01 %

4th Frequency 44.317 44.305 2.71e-02 %

5th Frequency 54.783 54.706 1.41e-01 %

6th Frequency 104.241 103.816 4.09e-01 %

7th Frequency 118.374 118.242 1.12e-01 %

8th Frequency 126.189 126.189 1.29e-05 %

9th Frequency 136.045 135.601 3.27e-01 %

10th Frequency 185.731 184.869 4.66e-01 %

Number of Dofs7 12288 54720

Table 6: Eigenfrequencies of I-section beam [Hz]

To offer researchers an entry point for code extensions, the

paper firstly explained the different modules of the framework

in detail by pointing out the class structure and its internal de-

pendencies. Secondly, the usage of the toolbox was explained

by means of different examples demonstrating how explicit and

voxel-based geometry models can be used for quasi-static and

eigenmode analyses within FCMLab.

At the current stage, the toolbox’s scope allows to apply the

Finite Cell Method in the context of one-, two-, and three-

dimensional linear-elasticity. However, the framework is ex-

plicitly designed to allow for easy extensibility. It is, therefore,

conceivable to broaden the applicability to non-linear problems

or to implement alternative integration schemes. Also, the de-

velopment of new approaches for imposing Dirichlet bound-

ary conditions on the non-conforming discretization would be

very interesting. We, therefore, explicitly invite interested re-

searchers to join our development group at http://fcmlab.

cie.bgu.tum.de/. Here, you can download the full source

code of FCMLab and find a detailed Getting Started guide ex-

plaining the installation of the framework and how to setup and

run your own examples.

Acknowledgements

The first two and the last author gratefully acknowledge the

financial support of the German Research Foundation (DFG)

under grant RA 624/19-1 and RA 624/15-2. D. Schillinger ac-

knowledges support from the Institute for Computational En-

gineering and Sciences (ICES) at the University of Texas at

Austin and the German Research Foundation (DFG) under

grant SCHI 1249/1-2.

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